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Abstract

We establish rigorous necessary analytical conditions for the existence of single-polarization single-mode (SPSM) bandwidths in index-guided microstructured waveguides (such as photonic-crystal fibers). These conditions allow us to categorize designs for SPSM waveguides into four strategies, at least one of which seems previously unexplored. Conversely, we obtain rigorous sufficient conditions for the existence of two cutoff-free index-guided modes in a wide variety of microstructured dielectric waveguides with arbitrary periodic claddings, based on the existence of a degenerate fundamental mode of the cladding (a degenerate light line). We show how such a degenerate light line, in turn, follows from the symmetry of the cladding.

Figures (8)

Schematics of various types of dielectric waveguides in which our theorem is applicable. Light propagates in the z direction (along which the structure is either uniform or periodic) and is confined in the xy direction by a higher-index core compared to the surrounding (homogeneous or periodic) cladding.

Example dispersion relation for a 3D rectangular waveguide in air of width a and height 0.4a (inset), showing the light cone, the light line, and fundamental and second (cutoff-free) guided modes, and higher-order modes with cutoffs.

. Example microstructured optical fiber cladding structures with three-fold, fourfold, six-fold and cylindrical rotation symmetries. Claddings with these symmetries are guaranteed to have a doubly-degenerate light line, at least in the long-wavelength limit for cases (a–c).

First two bands (Bloch modes) of a holey-fiber cladding (triangular lattice, period a, of radius 0.3a air holes in index-1.45 silica) plotted around the boundary of the irreducible Brillouin zone (inset). Each pair of bands is plotted for a fixed value of β:β = 0.001, 0.2, 0.5, and 1.0 in units of 2π/a. The bands are doubly degenerate, by symmetry, at the Γ point, and, because this is the lowest-frequency mode at each β, it is the (doubly degenerate) fundamental mode of the cladding and defines the light line.

Fig. 6. Dispersion relations of structures with an asymmetric cladding with no two-dimensional irreducible presentation. In both cases, the fundamental polarization is rigorously cutoff-free, with a single-polarization region below the second-mode cutoff at β = 0.82 and β = 0.5 respectively. Here, we plot the mode frequency ω as ωc-ω, where ωc is the light-line frequency–this difference is positive for a guided mode.

|D(1,2)c|2 field patterns of the two degenerate fundamental space-filling modes of the cladding of the structure in Fig. 8. We can utilize the asymmetry of the degenerate cladding mode and design an asymmetric core such that Eq. (2) is satisfied for one cladding mode but not the other.

Dispersion relation of a structure with an asymmetrical core in a symmetrical cladding of circular air holes (radius 0.47a in a hexagonal lattice with n = 1.87). The core is formed by two small cylinders of Δ=±0.18, respectively, shown in the inset as light and dark circles in the veins between two pairs of air holes. Here, we plot the mode frequency ω as ωc-ω, where ωc is the light-line frequency–this difference is positive for a guided mode.